Update:
My trailer has suspension but is neither braked nor plated, which means max trailer weight will be 750kg. It's probably safe to carry the lathe and I'd be happy for it to be used to do so, but it may be better to find a higher rated trailer.
The trailer is currently set up as a motorcycle trailer but I planned to put a wooden bed/box on it later. Not sure whether 18mm OSB would be man enough for the weight, I suspect not which means either load spreading with more wood or just get another trailer. I have a large number of unused roofing joists which would work well as load spreaders.
Either way the engine crane looks to be the way to move the lathe short distances. You'll need a lifting eye if the seller isn't providing one, which is probably M10 or M12. My crane is a 1t crane, i.e. 250kg lift at full reach, 500kg at mid reach and 1t at short reach. I can't find dimensional info but it'll be important to determine the gap between the pillars to see if the crane legs will fit under the lathe to lift it. It's also important to find out the lifting eye height, since if it's high the shorter reach of my 1t crane won't be enough to get it off the ground.
Having read about moving these lathes it looks like many hands make light work. Obviously doing the move outside 'normal' working hours will mean that we can get more hands. Might also be worth building/borrowing/buying some very high strength moving dollies so that the lathe can be moved into the space without the hassle of moving an engine crane around.
Let me know if you need me…
Stuart